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On the Loop Homology of a Certain Complex of RNA Structures

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  • Thomas J. X. Li

    (Biocomplexity Institute and Initiative, University of Virginia, Charlottesville, VA 22904-4298, USA
    Physical Address: Biocomplexity Institute, Town Center Four, 994 Research Park Boulevard, Charlottesville, VA 22911, USA.
    Mailing Address: Biocomplexity Institute, P.O. Box 400298, Charlottesville, VA 22904-4298, USA.)

  • Christian M. Reidys

    (Biocomplexity Institute and Initiative, University of Virginia, Charlottesville, VA 22904-4298, USA
    Mathematics Department, University of Virginia, Charlottesville, VA 22904-4137, USA
    Physical Address: Biocomplexity Institute, Town Center Four, 994 Research Park Boulevard, Charlottesville, VA 22911, USA.
    Mailing Address: Biocomplexity Institute, P.O. Box 400298, Charlottesville, VA 22904-4298, USA.)

Abstract

In this paper, we establish a topological framework of τ -structures to quantify the evolutionary transitions between two RNA sequence–structure pairs. τ -structures developed here consist of a pair of RNA secondary structures together with a non-crossing partial matching between the two backbones. The loop complex of a τ -structure captures the intersections of loops in both secondary structures. We compute the loop homology of τ -structures. We show that only the zeroth, first and second homology groups are free. In particular, we prove that the rank of the second homology group equals the number γ of certain arc-components in a τ -structure and that the rank of the first homology is given by γ − χ + 1 , where χ is the Euler characteristic of the loop complex.

Suggested Citation

  • Thomas J. X. Li & Christian M. Reidys, 2021. "On the Loop Homology of a Certain Complex of RNA Structures," Mathematics, MDPI, vol. 9(15), pages 1-22, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1749-:d:600797
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